By Stefan Liebscher
Targeted at mathematicians having no less than a easy familiarity with classical bifurcation concept, this monograph offers a scientific type and research of bifurcations with no parameters in dynamical platforms. even if the tools and ideas are in brief brought, a previous wisdom of center-manifold discounts and normal-form calculations may help the reader to understand the presentation. Bifurcations with out parameters happen alongside manifolds of equilibria, at issues the place common hyperbolicity of the manifold is violated. the overall conception, illustrated by way of many functions, goals at a geometrical knowing of the neighborhood dynamics close to the bifurcation points.
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Additional info for Bifurcation without Parameters
We shall construct a particular source g with a straight line of equilibria. 0/. 7) holds. 8) is translated into a non-degeneracy condition on f . 7) imposed by the structure of one conservation law and two balance laws. 0/ D S 1 S D diag. 10) Here we have normalized the imaginary part of the Hopf eigenvalue to one. 8) onto Hopf normal form. 0/ which is equivalent to S T e3 ? 0; 0; 1/T denotes the third standard unit vector. 10). a2 ; a4 / 2 R2 . 12) we can choose a basis of R3 by: 0 1 1 v1 WD @ 0 A ; 0 0 1 0 v2 WD @ a1 A ; a3 1 v3 WD @ a2 A ; a4 v 2 ; v 3 2 .
1a. u˙ 1 ; 0; : : : ; 0/ on opposite sides of the Hopf point u D 0. u/ is real analytic near u D 0, then the two-dimensional strong stable and strong unstable manifolds of u˙ within the center manifold intersect at an angle which possesses an exponentially small upper bound in terms of ju˙ j. See Fig. 1b. The formulation of the assumptions of the above theorem can be simplified: we first restrict to the three dimensional center manifold and assume that this manifold is flat. Then we take coordinates in direction of the real, generalized eigenvectors of the linearization at the Hopf point.
See Fig. 1b. The formulation of the assumptions of the above theorem can be simplified: we first restrict to the three dimensional center manifold and assume that this manifold is flat. Then we take coordinates in direction of the real, generalized eigenvectors of the linearization at the Hopf point. 0; y/ Á 0. 0; 0/ > 0. 0; 0/ ¤ 0; D @2x1 C @2x2 . The first condition is our structural assumption, (ii) describes our bifurcation point, and (iii,iv) are non-degeneracy assumptions fulfilled generically.